hw6_solutions

# hw6_solutions - Homework 6 Solutions Math 471 Fall 2006...

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Homework 6 Solutions Math 471, Fall 2006 Assigned: Friday, October 20, 2006 Due: Friday, October 27, 2006 Include a cover page Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Condition Numbers) P. 187 #1, 2 Part (a): κ ( AB ) = || AB || · || B - 1 A - 1 || ≤ || A || · || A - 1 || · || B || · || B - 1 || = κ ( A ) κ ( B ). Part (b): κ ( αA ) = || αA || · || 1 α A - 1 || = | α | | α | || A || · || A - 1 || = κ ( A ). (2) (Error Estimates) P. 188 #7c, 8c 1 - 1 - 1 0 1 - 1 0 0 1 x = 0 2 0 , x = 2 2 0 , ˜ x = 1 . 9000 2 . 1000 - 0 . 1000 || x || = 2 , e = x - ˜ x = 0 . 1000 - 0 . 1000 0 . 1000 , || e || = 0 . 1 r = b - A ˜ x = 0 . 1000 - 0 . 2000 0 . 1000 , || r || = 0 . 2 , || b || = 2 || A || = 3 , || A - 1 || = 4 , κ ( A ) = 12 || e || || x || = 0 . 1 2 = κ ( A ) 24 || r || || b || κ ( A ) || r || || b || = 12 0 . 2 2 = 12 · (0 . 1) (3) (Positive Definite and Strictly Diagonally Dominant Matrices) P. 220 #3 Part (a) : A matrix is positive definite if and only if the determinants of the principal submatrices are all positive. Δ 1 = a > 0 , Δ 2 = 4 a - 1 > a > 1 4 Δ 3 = det( A ) = 19 a - 5 > 0 a > 5 19 The intersection of these three intervals is a > 5 / 19 which extends the interval a > 1 we obtained using the definition of the positive definite matrix. Therefore, if a > 5 / 19, then A is positive definite. Part (b) : The matrix A is strictly diagonally dominant if 1

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2 | a ii | > n X j =1 , j 6 = i | a ij | Then | a | > 1 a < - 1 or a > 1 4 > 2 OK 5 > 1 OK Therefore, A
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